Question

How is the variational principle applied?

Question 3 options:

	a) To find an upper bound for the ground-state kinetic energy
	b) To find an upper bound for the ground-state potential energy
	c) To find a lower bound for the ground-state kinetic energy
	d) To find a lower bound for the ground-state potential energy
	e) To find an upper bound for the ground-state total energy

Answer 1

Answer is e) To find an upper bound for the ground-state total energy

Variational principle helps us to get an approximate upper bound on the total ground state energy of a given system. What we do for this is to choose a trial wave function with certain parameters. And then for given Hamilton operator, we try to find values of these parameters for which energy eigenvalue is the smallest possible value. And that gives us an approximate upper bound on the ground state total energy. This approximation can always be improved to get more precise value of ground state total energy by choosing proper trial function. If somehow you choose a wave function that looks more like the original wave function then you will get better results.

In this question one is tempted to choose the options a) & b) as well, since we can always write Hamilton operator as and perform the same calculations and expect to get the good results for KE and PE. But here's a catch, we know that ground state total energy is always the lowest in any system, because it is ground state right :) But do we really know that the ground state kinetic energy has the lowest value? Or ground state potential energy has the lowest value? It could be always possible that ground state kinetic energy for a given system is very high but its ground state potential energy is very negative and then together they make a lowest ground state total energy. So applying the same method to find KE and PE may not be always possible.

=======================================================

If you still have any doubts, please let me know in the comment section. All the best for your exams :)